2,804 research outputs found
Sharp H\"older continuity of tangent cones for spaces with a lower Ricci curvature bound and applications
We prove a new kind of estimate that holds on any manifold with lower Ricci
bounds. It relates the geometry of two small balls with the same radius,
potentially far apart, but centered in the interior of a common minimizing
geodesic. It reveals new, previously unknown, properties that all generalized
spaces with a lower Ricci curvature bound must have and it has a number of
applications.
This new kind of estimate asserts that the geometry of small balls along any
minimizing geodesic changes in a H\"older continuous way with a constant
depending on the lower bound for the Ricci curvature, the dimension of the
manifold, and the distance to the end points of the geodesic. We give examples
that show that the H\"older exponent, along with essentially all the other
consequences that we show follow from this estimate, are sharp. The unified
theme for all of these applications is convexity.
Among the applications is that the regular set is convex for any
non-collapsed limit of Einstein metrics. In the general case of potentially
collapsed limits of manifolds with just a lower Ricci curvature bound we show
that the regular set is weakly convex and convex, that is almost every
pair of points can be connected by a minimizing geodesic whose interior is
contained in the regular set. We also show two conjectures of Cheeger-Colding.
One of these asserts that the isometry group of any, even collapsed, limit of
manifolds with a uniform lower Ricci curvature bound is a Lie group; the key
point for this is to rule out small subgroups. The other asserts that the
dimension of any limit space is the same everywhere. Finally, we show that a
Reifenberg type property holds for collapsed limits and discuss why this
indicate further regularity of manifolds and spaces with Ricci curvature
bounds.Comment: 48 page
Tangential dimensions I. Metric spaces
Pointwise tangential dimensions are introduced for metric spaces. Under
regularity conditions, the upper, resp. lower, tangential dimensions of X at x
can be defined as the supremum, resp. infimum, of box dimensions of the tangent
sets, a la Gromov, of X at x. Our main purpose is that of introducing a tool
which is very sensitive to the "multifractal behaviour at a point" of a set,
namely which is able to detect the "oscillations" of the dimension at a given
point. In particular we exhibit examples where upper and lower tangential
dimensions differ, even when the local upper and lower box dimensions coincide.
Tangential dimensions can be considered as the classical analogue of the
tangential dimensions for spectral triples introduced in math.OA/0202108 and
math.OA/0404295, in the framework of Alain Connes' noncommutative geometry.Comment: 18 pages, 4 figures. This version corresponds to the first part of
v1, the second part being now included in math.FA/040517
Dimensional flow and fuzziness in quantum gravity: emergence of stochastic spacetime
We show that the uncertainty in distance and time measurements found by the
heuristic combination of quantum mechanics and general relativity is reproduced
in a purely classical and flat multi-fractal spacetime whose geometry changes
with the probed scale (dimensional flow) and has non-zero imaginary dimension,
corresponding to a discrete scale invariance at short distances. Thus,
dimensional flow can manifest itself as an intrinsic measurement uncertainty
and, conversely, measurement-uncertainty estimates are generally valid because
they rely on this universal property of quantum geometries. These general
results affect multi-fractional theories, a recent proposal related to quantum
gravity, in two ways: they can fix two parameters previously left free (in
particular, the value of the spacetime dimension at short scales) and point
towards a reinterpretation of the ultraviolet structure of geometry as a
stochastic foam or fuzziness. This is also confirmed by a correspondence we
establish between Nottale scale relativity and the stochastic geometry of
multi-fractional models.Comment: 25 pages. v2: minor typos corrected, references adde
The multiplicative coalescent, inhomogeneous continuum random trees, and new universality classes for critical random graphs
One major open conjecture in the area of critical random graphs, formulated
by statistical physicists, and supported by a large amount of numerical
evidence over the last decade [23, 24, 28, 63] is as follows: for a wide array
of random graph models with degree exponent , distances between
typical points both within maximal components in the critical regime as well as
on the minimal spanning tree on the giant component in the supercritical regime
scale like .
In this paper we study the metric space structure of maximal components of
the multiplicative coalescent, in the regime where the sizes converge to
excursions of L\'evy processes "without replacement" [10], yielding a
completely new class of limiting random metric spaces. A by-product of the
analysis yields the continuum scaling limit of one fundamental class of random
graph models with degree exponent where edges are rescaled by
yielding the first rigorous proof of the above
conjecture. The limits in this case are compact "tree-like" random fractals
with finite fractal dimensions and with a dense collection of hubs (infinite
degree vertices) a finite number of which are identified with leaves to form
shortcuts. In a special case, we show that the Minkowski dimension of the
limiting spaces equal a.s., in stark contrast to the
Erd\H{o}s-R\'{e}nyi scaling limit whose Minkowski dimension is 2 a.s. It is
generally believed that dynamic versions of a number of fundamental random
graph models, as one moves from the barely subcritical to the critical regime
can be approximated by the multiplicative coalescent. In work in progress, the
general theory developed in this paper is used to prove analogous limit results
for other random graph models with degree exponent .Comment: 71 pages, 5 figures, To appear in Probability Theory and Related
Field
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